Figure REF  illustrates the effect of trend filtering on a couple \(\mathbf {v}\)  components. One important difference compared to previous denoising approaches [1]}, [2]} is that the TF model is more flexible than the sparse autoregressive model that is typically used to denoise calcium imaging data: the TF model does not require the estimation of any sparsity penalties or autoregressive coefficients, and can handle a mixture of positive and negative fluctuations, while the sparse nonnegative autoregressive model can not (by construction). This is important in this context since each component in \(\mathbf {V}\)  can include multiple cellular components (potentially with different timescales), mixed with both negative and positive weights.
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